accircuits2 - Complex Numbers and AC Circuits We pretend...

Complex Numbers and AC Circuits We pretend that there is an (imaginary) number, i , which, multiplied by itself, equals &1: 1 − ≡ i (1) A so-called complex number, z = x + iy , has both, a real part (Re( z ) = x ) and an imaginary part (Im( z ) = y ). The complex conjugate z * of z one obtains by flipping the sign of all terms with an i in them, i.e., z * = x & iy . Leonhard Euler (1707 & 1783) discovered the relation, which relates complex numbers to the (periodic) trigonometric functions (and which is the reason why we are doing what we are doing!): α sin cos ⋅ + = i e i (2) Consider the voltage U(t) that appears somewhere in some AC circuit. This is a periodic function, so it makes sense to try to write this as t i e U t U ω ⋅ =0 ) ( (3) This is a complex number, but instead of specifying its real and imaginary parts, we specify its amplitude U0 and its phase ω t . Here, ω is the angular frequency, which is related to the frequency f by ω = 2 π f . When we have any complex number

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